Particle dynamics on a cosine path.

[gilgtc]

Hello, I have attached the problem that I am referring to.

Basically I have a particle traveling on a path defined by a cosine function and I want the equations of motion with respect to the inertial system, if the particle is under the influence of friction and gravity.

I have calculated the equation of motion with respect to the moving coordinate system s but I am not sure how to relate that to x1 and x2 (the inertial system) which is what I ultimately need.

Your help is very much appreciated.

Thank you,

gilgtc

 

[RTW69]

The bob of a pendulum started from height "h" will exhibit motion defined by a cosine function. Because of friction in the piviot the pendulum will not oscillate indefinitely rather it will exhibit damped harmonic motion. I believe this is a classic Differential equation problem.

 

[Moetasim]

Hello gilgtc,

Thank you for reaching out to me with your question. It sounds like you are working on a fascinating problem involving particle dynamics on a cosine path. To help you out, I will provide some guidance on how to approach this problem and how to relate the equations of motion in the moving coordinate system to the inertial system.

First, let's define the problem. You have a particle traveling on a path defined by a cosine function, which we can represent as y = A cos(ωt), where A is the amplitude and ω is the angular frequency. This path can also be described in terms of x and y coordinates as x = A cos(ωt) and y = 0.

Now, let's consider the forces acting on the particle. You mentioned that the particle is under the influence of friction and gravity. We can represent these forces as Ffr and Fg, respectively. Ffr will act in the opposite direction of the particle's motion and Fg will act downwards.

Next, we need to determine the equations of motion with respect to the inertial system, which we can do by using Newton's second law, F = ma. In this case, the forces acting on the particle are Ffr and Fg, so we can write:

F = Ffr + Fg = ma

Since we know that the acceleration is the second derivative of the position with respect to time, we can rewrite this equation as:

m(d^2x/dt^2) = Ffr + Fg

To solve this equation, we need to relate the position x to the time t. This can be done by using the cosine function, as we know that x = A cos(ωt). We can also relate the velocity v to the time t by taking the derivative of x with respect to t:

v = dx/dt = -Aω sin(ωt)

Using these relationships, we can rewrite the equation of motion as:

m(d^2/dt^2)(A cos(ωt)) = Ffr + Fg

Now, we need to relate this equation to the equations of motion in the moving coordinate system s. To do this, we can use the chain rule:

(d/dt) = (d/ds)(ds/dt)

Since we know that s = x - vt, we can rewrite the equation as:

m(d^2/ds^2)(A cos(ωt